<span class="mathjax-formula">$\protect \mathbf{BV}$</span> solutions of rate independent variational inequalities

BV solutions of rate independent variational inequalities

VINCENZORECUPERO

Abstract. We prove a theorem providing a geometric characterization ofBV continuous vector rate independent operators. We apply this theorem to rate inde- pendent evolution variational inequalities and deduce new continuity properties of their solution operator: the vectorial play operator.

Mathematics Subject Classification (2010):49J40 (primary); 47J35, 74C15, 34C55, 26A45 (secondary).

1.

Introduction

In several mathematical models of elastoplasticity, the nonlinear dependence be- tween deformation and stress tensors is described by means of the following evolu- tion variational inequality. Let

H

be a real Hilbert space with inner product·,·and

Z

⊆

H

be a closed convex subset containing 0. GivenT >0 andu:[0,T]−→

H

^, find y:[0,T]−→

H

^{such that}

u(t)−y(t)−z,y(t) ≥0 ∀z ∈

Z

, t ∈[0,T], (1.1) whereydenotes the time derivative ofy. The references [14,15,20] contain surveys of the physical models described by (1.1). The special one dimensional case

H

=

R

has been deeply studied by several authors: we refer to the monographs [6, 12, 19, 27].

Inequality (1.1) can be solved by using classical tools from the theory of evo- lution equations governed by maximal monotone operators. In particular it is well known that if u ∈ W^1,1(0,T;

H

) then there exists a unique y ∈ W^1,1(0,T;

H

) satisfying (1.1) and the initial condition

u(0)−y(0)=z₀, (1.2)

where z₀ ∈

Z

is fixed. The resulting solution operator P : W^1,1(0,T;

H

) −→

W^1,1(0,T;

H

)is usually called (vector)play operator. The suggestive termsinput andoutputare used foru andvrespectively. Regarding problem (1.1)-(1.2) there are two important issues to be considered. First of all the continuity of the solution Received November 14, 2009; accepted in revised form March 4, 2010.

operatorP :u −→ ywith respect to different topologies. Secondly the extension of such operator to classes of functions more general than W^1,1(0,T;

H

). Both questions have an applicative relevance since the continuity of the extension ensures robustness of the model and applicability of mathematical tools including numerical simulation.

It is well-known that the operatorPis continuous onW^1,1(0,T;

H

)endowed with its natural topology: this was proved in [13, Proposition 3.1] in the finite di- mensional case, whereas for general

H

it is proved in [14, Theorem 3.12, page 34].

The continuity with respect to the topology of uniform convergence is proved in [14, Corollary 3.8, page 32]. As far as the extension ofPis concerned, it seems that the first answer to this question in the infinite dimensional case can be found in [14]. In that book the play operator is extended to the spaceBV(0,T;

H

^)∩C(0,T;

H

^{). In} order to do this, the variational inequality (1.1) is replaced by the integral inequality

_T

0

u(t)−y(t)−z(t),dy(t)

≥0 ∀z∈C([0,T];

Z

). (1.3) Here the integral is meant in the sense of Riemann-Stieltjes. In [14] the problem (1.3)–(1.2) is first solved for step functions, then a solution for continuous BV inputs is found by ana priori estimates-limit procedure. In [14] it is also proved the continuity with respect to the topology of uniform convergence and it is shown that this extension ofPis continuous inBV(0,T;

H

)∩C(0,T;

H

)endowed with the strict metric, provided

Z

is bounded and its boundary satisfies suitable smoothness conditions, the general case being left as an open problem. Let us recall that the strict metricis defined by

d_s(u, v):= u−v_L¹+ |V(u,[0,T])−V(v,[0,T])|, (1.4) where V(u,[0,T])is the variation ofuon [0,T]. This is a natural metric on BV be- cause every functionuof bounded variation admits a sequence of smooth functions u_nsuch thatd_s(u_n,u)→0 asngoes to infinity.

In the paper [16] the play operator is further extended to the space of possibly discontinuous functions of bounded variation. In that paper the integral in (1.3) is understood in the Young sense and the continuity with respect to the topology of uniform convergence is proved. The continuity with respect to the strict conver- gence ofBV is left as an open question.

In the present paper we address the issue of BV-continuity by studying the problem of the extension of a generalrate independent operator: indeed the play operatorPisrate independent,i.e.

P(u◦φ)=P(u)◦φ (1.5)

wheneveru∈W^1,p(0,T;

H

)andφ:[0,T]−→[0,T] is an increasing surjective Lipschitz reparametrization. Thus we study when a general rate independent oper- atorR, acting on the space of Lipschitz mappings, can be continuously extended to

allBV(0,T;

H

). In our main theorem we prove that such extension exists if and only ifRislocally isotone,i.e.

V(u,[c,d])=u(d)−u(c)_H=⇒V(R(u),[c,d])=R(u)(d)−R(u)(c)_H (1.6) wheneveruis Lipschitz and [c,d] is a subinterval of [0,T]. Moreover this exten- sion is unique if we identify functions which are equal almost everywhere. Con- dition (1.6) has the advantage of a clear geometrical meaning that can be easily applied to the play operator and translated in terms of the convex set

Z

^{. It turns} out thatPcan be continuously extended toBV(0,T;

H

)if and only if either

Z

^{is a} vector subspace or

Z

= {x ∈

H

: −α≤ f,x ≤β} (1.7)

for some f ∈

H

^{{0}andα, β ∈ [0,}∞]. Therefore in many simple cases (e.g.

Z

is a cylinder or a ball for dim(

H

^{) >} 1) the operatorP cannot be continuously extended toBV. However, as a by-product of the proof, we obtain thatPcan always be continuously extended toBV(0,T;

H

^)∩C(0,T;

H

^{)for every}

Z

. Therefore we extend the result of [14] where the continuity is proved only for smooth

Z

^.

The scalar case was dealt in the papers [21, 22] where we proved that aBV- continuous scalar rate independent operatorR:W^1,∞(0,T;

R

^{)−→W1,∞(0,T;}

R

⁾ can be continuously extended toBV(0,T;

R

⁾(in a unique manner) if and only if it is locally isotone. When

H

=

R

, local isotonicity is a very natural generalization of local monotonicity, well-known in hysteresis: a scalar operatorRis called locally monotone if

u increasing (resp. decreasing) on [c,d]=⇒R(u)increasing (resp. decreasing) on [c,d]

whenever [c,d] is a subinterval of [0,T]. In this special case every convex set

Z

is an interval andP is locally monotone, thereforePcan always be continuously extended toBV(0,T;

R

). In applications, locally monotonicity is verified in many particular cases, therefore the result applies to a wide class of concrete rate inde- pendent operators (cf.[21, Section 5]).

Let us also observe that our procedure will yield a representation formula for the extensionPofP(and in general for a rate independent operatorR). Indeed we prove that

P(u)=P(u)◦u, (1.8)

where

u(t)= T

V(u,[0,T])V(u,[0,t]) (1.9) anduis a Lipshitz map such that

u=u◦u. (1.10)

Even ifPcannot be continuosly extended to all ofBV(0,T;

H

), we prove thatP has a good continuity property, namely P(u_n) → P(u)in L¹(0,T;

H

)whenever

u_n converges strictly tou. This property suggests thatP(u)can be defined to be a generalized solution of (1.1)-(1.2) whenu is of bounded variation. We show that this notion of solution does not coincide with the one proposed in [16], indeed it solves a variational inequality similar to (1.3), but containing an extra term due to the jumps ofu. A comparison between this two notions of solution is given.

Now we give a brief plan of the paper. In the next section we recall the main definitions and notations about vector valued functions of bounded variation. In Sections 3 and 4 we state precisely the main results and we present their proofs. In Section 5 we apply the abstract results to rate independent variational inequalities.

Finally in the Appendix we prove some technical results about

H

^{-valuedBV maps} and convex sets in a Hilbert space.

ACKNOWLEDGEMENTS. The author is grateful to P. Krejˇc´ı and G. Savar´e for stimulating discussions and useful suggestions.

2.

Preliminaries

2.1. List of notation

In the paper we will use the following notation.

· B^A, set of functions defined on a set Awith values in a setB.

·

P

(S), power set of a setS.

·

N

, set of strictly positive integer numbers{1,2, . . .}.

· χS, characteristic function of a setS:χS(t)=1 ift ∈SandχS(t)=0 ift ∈ S.

· S, closure of a subsetS⊆T, withT topological space.

· S, interior of a subset˚ S⊆T, withT topological space.

· f(t−) := lim_st f(s), f(t+) := lim_st f(s), with f ∈ T^S, T topological space,S⊆

R

^.

· Cont(f), continuity set of f ∈T^S, withS,T topological spaces.

· Discont(f)= S

^{Cont(f), with f ∈TS andS,T} topological spaces.

· f_∞ := sup{f(s)_X : s ∈ S}, with f ∈

X

^{S, S set,} (

X

, · _X)Banach space.

· Lip(I;

X

),

X

-valued Lipschitz continuous functions defined onI, with I ⊆

R

interval,

X

Banach space.

· Lip(f), Lipschitz constant of f ∈

X

^{I, withI ⊆}

R

^interval,

X

Banach space.

·

X

, topological dual space of a Banach space

X

^.

· x_n x, weak convergence: f(x_n)→ f(x)for all f ∈

X

^{, withx}n,x ∈

X

^,

X

Banach space.

· f_n ^∗ f, weak* convergence: f_n(x)→ f(x)for allx ∈

X

^{, with fn, f ∈}

X

^,

X

Banach space.

· seg[x,y] := {(1−λ)x+λy : λ∈[0,1]}, segment joiningxandyin a Hilbert space.

· Proj_Z, projection operator on a closed convex set

Z

in a Hilbert space.

·

B

(T), family of Borel sets ofT,T topological space.

· L¹(µ,T;

X

^)=L1(µ;

X

^{), space ofµ}-integrable

X

-valued maps, withµposi- tive measure onT.

· L¹, one dimensional Lebesgue measure, L¹(I;

X

) := L¹(L¹,I;

X

), I ⊆

R

interval.

Let us emphasize that we do not identify two functions defined on the real line which are equalL¹-almost everywhere (L¹-a.e.). Moreover throughout the paper we assume that

I :=]a,b[, −∞ ≤a<b≤ ∞, (2.1) the open interval in

R

with endpointsa,b, and

H

is a real Hilbert space with inner product(x,y) −→ x,y

x_H:= x,x^1/2. (2.2)

2.2. Pointwise and essential variations

In this subsection

X

denotes a Banach space with norm · _X. We collect the main definitions and results concerning Banach valued functions with bounded pointwise variation. All the results are standard in the real case, however we give proofs whenever we are not able to provide a reference for the vector case.

Definition 2.1. If J is a subinterval of I, the symbol St(J;

X

)denotes the set of

X

-valued step maps on J,i.e. maps f : J −→

X

^{such that J} can be partitioned into a finite number of (possibly degenerate) intervals J₁, . . . ,J_mand f is constant on each J_j for j = 1, . . . ,m. A function f : J −→

X

^{is called}regulated on J if at each pointt ∈ J there exist f(t−)and f(t+)in

X

, with the convention that f(t−):= f(t)(respectively f(t+):= f(t)) ift ∈J andt is the right (resectively the left) endpoint of J. We denote byReg(J;

X

)the set of regulated maps onJ.

Every f ∈Reg(J;

X

)is locally the uniform limit of a sequence f_n ∈St(J;

X

) (cf.,e.g., [3, Theorem 3, Section 2.1]), hence f isL¹-measurable, the set{t ∈ J : f(t−) = f(t+)}is at most countable, and if J is compact then f is bounded.

Of course every monotone real function is regulated. In this regard we warn the reader about the terminology: by anincreasing functiononJ, we mean a function f : J −→

R

^{such that(f(t1)− f(t2))(t1−t2)≥}0 for everyt₁,t₂∈J. Same con- vention is adopted for the termdecreasing. Finally f ismonotoneif it is increasing or if it is decreasing.

Definition 2.2. We recall that asubdivisionof a nondegenerate subinterval J ⊆ I is a family(s_j)^m_j=0,m ∈

N

, with the property thats₀ < · · · < s_m ands_j ∈ J for j = 0, . . . ,m. The set of all subdivisions of J is indicated by

S

(J). If f ∈

X

^I and

s

=(s_j)^m_j=0∈

S

(J), thevariation of f with respect to

s

is defined by

V(f,

s

):=^m

j=1

f(s_j)− f(s_j−1)_X.

If J is nondegenerate thepointwise variation of u on J is defined by V_p(f,J):=sup

V(f,

s

_{) :}

s

_∈

S

₍J) ,

otherwise we set V_p(f,J)=0. We defineBV_p(I;

X

):={f∈

X

^I : V_p(f,I)<∞}. If f ∈

X

^{I, V}p(f,I) <∞, andt₀ ∈ I, the inequalityf(t)_X ≤V_p(f,I)+ f(t₀)_X yields the boundedness of f. Moreover it is well known there exist (in

X

⁾ lim_t→infI+ f(t), lim_t→supI− f(t), f(t+), and f(t−)for everyt ∈I. In particular f is regulated,L¹-measurable, and Discont(f)is at most countable. We can define the maps f₋, f₊ ∈

X

^{I by setting}

f₋(t):= f(t−), f₊(t):= f(t+), t ∈I. (2.3) It is easy to check that V_p(f₊,I)=V_p(f₋,I). Let us observe that ifg₁,g₂ ∈

X

^I are two functions in the sameL¹-equivalence class and V_p(g_j,I) <∞, j =1,2, then everyt ∈I is a left Lebesgue point ofg_j, hence

(g₁)₋(t)= lim

h0

1 h

_t

t−h

g₁(s)ds= lim

h0

1 h

_t

t−h

g₂(s)ds=(g₂)₋(t).

In the same manner we see that(g₁)₊=(g₂)₊. This remark allows us to formulate the following:

Definition 2.3. Let f ∈

X

^I be given. If there is noL¹-representativegof f such that V_p(g,I) <∞, we set V_e(f,I):= ∞. Otherwise ifg∈

X

^I is such that f =g L¹-a.e. inI and V_p(g,I) <∞, we set

V_e(f,I):=V_p(g₋,I) (=V_p(g₊,I)), (2.4) whereg₋is defined in (2.3). The real extended number V_e(f,I)is calledessential variation of f.

Now let f ∈

X

^I be left-continuous, then Discont(f) is at most countable (proof: to every t ∈ Discont(f)associate a triple (p,q,r) ∈

Q

³ such that 0 <

p<lim sup_τ→tf(τ)− f(t)_X,f(s)− f(t)_X < pwheneverq<s <t, and lim sup_τ→sf(τ)− f(s)_X > pwhenever t < s < r; from the left continuity

it follows that the correspondence t −→ (p,q,r) is one-to-one). Therefore if

s

_{= (}s_j)^m_j=1is a subdivision of I and f is left-continuous, then for every ε > 0 we can find another subdivision

t

= (t_j)^m_j=1such thatt_j ∈Cont(f),t_j < s_j, and f(t_j)−f(s_j)_X < ε/(4m)for j =1, . . . ,m. Hence V(f,

s

)+ε/2≤V(f,

t

)+ε, thus we have proved the following:

Lemma 2.4. If f : I −→

X

is left-continuous, then V_p(f,I)=sup

V(f,

s

) :

s

=(s_j)∈

S

(I), s_j ∈Cont(f) .

Let us notice that if f, fn ∈

X

^{I and fn}(t) → f(t)for everyt ∈ Cont(f), then V(f_n,

s

)→V(f,

s

)for every

s

∈

S

(I). Hence thanks to Lemma 2.4 we have the following:

Corollary 2.5. Assume that f, f_n ∈

X

^{I and f}n(t)→ f(t)for every t ∈Cont(f). ThenV_e(f,I)≤lim inf_n→∞V_e(f_n,I).

2.3. Vector Stieltjes measures

Now we recall the connection between functions with bounded variation andBorel vector measureson the real line,i.e.mapsµ:

B

(I)−→

X

^{such that}µ( ^∞_n=1B_n)=

_∞

n=1µ(B_n)whenever(B_n)is a sequence of mutually disjoint Borel subsets ofI. Let us also recall that ifµ:

B

(I)−→

X

is a vector measure, thenµ:

B

(I)−→

[0,∞] is defined by



µ(B):=sup _∞

n=1

µ(B_n)_X : B=^∞

n=1

B_n,B_n∈

B

(I), B_h∩B_k=

∅

^ifh=k

. The mapµis a positive measure which is calledtotal variation ofµand we set

A(µ):= {t ∈I : µ({t})=0}, (2.5) the set of atoms of µ. The vector measureµis called with bounded variation if



µ(I) <∞(see,e.g., [8, Chapter I, Section 3.]). In this case the equalityµ :=



µ(I)defines a norm on the space of measures with bounded variation. Let us recall the following proposition whose proof can be found in [8, Theorem 1, section III.17.2, page 358]:

Theorem 2.6. If f ∈

X

^{I andV}p(f,I) < ∞ then there exists a unique vector measureµf :

B

(I)−→

X

such that for every c,d ∈I with c<d we have

µf(]c,d[)= f(d−)− f(c+), µf([c,d])= f(d+)− f(c−), (2.6) µf([c,d[)= f(d−)− f(c−), µf(]c,d])= f(d+)− f(c+). (2.7) Moreoverµf is with bounded variation and if f₋ : I −→

X

is defined by(2.3), thenµf =µf₋. Vice versa ifµ:

B

(I)−→

X

is a vector measure with bounded variation, then the map f_µ : I −→

X

defined by f_µ(t):= µ(]a,t[)is such that V_p(f_µ,I) <∞andµf_µ =µ.

Usuallyµf is calledthe Lebesgue-Stieltjes measure associated with f. Observe that from Theorem 2.6 it follows that µf(I) = lim_t→supI− f(t) − lim_t→infI+ f(t)and that µf({t}) = f(t+)− f(t−) for every t ∈ I. Now we recall the characterization of the total variation of µf (see [8, Remark 5, Section III.17.2, page 362]):

Proposition 2.7. Let f : I −→

X

be such thatV_p(f,I) <∞and let f₋: I −→

X

be defined by(2.3). Define V_f : I −→ ]0,∞]by V_f(t):=V_p(f₋,]a,t[), that is the pointwise variation of f₋on]a,t[. Thenµf



=µV_f

It follows that if f ∈

X

^{I, V}p(f,I)andJ is an open subinterval ofI, then



µf



(J)=V_p(f₋,J)=V_e(f,J). (2.8)

2.4. Integrals with respect to vector measures

Let

X

j, j = 1,2,3, be Banach spaces with norms · _Xj and let

X

1×

X

2 −→

X

3:(x₁,x₂) −→x₁•x₂be a bilinear form such thatx₁•x_2X3≤ x_1X1x_2X2 for every x_j ∈

X

j, j = 1,2. Assume thatµ :

B

(I)−→

X

2is a vector measure with bounded variation. Let f :

X

₁^{I be a}step map with respect toµ,i.e.there exist f₁, . . . , f_m ∈

X

1and A₁, . . . ,A_m ∈

B

(I)mutually disjoint such thatµ(A_j) <

∞for every j and

f = m

j=1

χA_j f_j.

The set of step maps with respect toµis denoted bySt(µ;

X

¹⁾and the integral of f is the vector defined by

I

f •dµ:=

m j=1

f_j •µ(A_j)∈

X

³.

It can be proved that the map St(µ;

X

1) −→

X

3 associating to every f the integral

I f •dµis linear and continuous whenSt(µ;

X

1)is endowed with the L¹-semimetricf − g_L¹₍^µ;X1) :=

If −g_X1dµ. Therefore it admits a unique continuous extension I_µ:L¹(µ;

X

1)−→

X

3and we set

I

f •dµ:=I_µ(f), f ∈L¹(µ;

X

^1).

The following fundamental inequality holds:

I

f •dµ X3

≤

I

f_X1dµ. (2.9)

We will use the previous integral in two particular cases, namely when a)

X

¹ =

R

^,

X

²=

X

³ =

H

^,λ•x :=λx (

I f •dµ=

I f dµ, integral of a real function with respect to a vector measure);

b)

X

^{1 =}

X

^{2 =}

H

^,

X

^{3 =}

R

^{,x1•x2 := x1,x2(}I f •dµ=

If,dµ, integral of a vector function with respect to a vector measure).

2.5. Maps whose derivative is a measure

Now we are going to present a brief summary of facts about functions of bounded variation with values in

H

. We adopt the notations of [2], which is our main refer- ence for the finite dimensional case.

Definition 2.8. A mapu ∈ L¹(I;

H

)is calledof bounded variation (on I) if its distributional derivative is a measure with bounded variation, i.e. if there exists a measure Du:

B

(I)−→

H

^{such thatDu(I}) <∞and

−

Iϕ(t)u(t)dt =

IϕdDu ∀ϕ∈C¹_c(I;

R

).

We set A(u):=A(Du)and the space of maps of bounded variation onI is denoted byBV(I;

H

).

Proposition 2.9. Assume that u ∈ BV(I;

H

) and define v ∈

H

^{I by} v(t) :=

Du(]a,t[). Then v is left-continuous, V_p(v,I) < ∞, and Du = µ_v = Dv. Moreover there exists a unique u_a∈

H

^{such that}

u(t)=u_a+Du(]a,t[) forL¹-a.e. t∈ I. (2.10) We have

V_e(u,I)=V_p(v,I) <∞.

Proof. The left continuity ofvis a straightforward consequence of the continuity of measures. It is easy to check that V_p(v,I)≤Du(I) < ∞. The last part of Theorem 2.6 yieldsµ_v= Du. Now takeϕ ∈C¹_c(I;

R

⁾. Thanks to Lemma A.1 of Section A.1 in the Appendix we have

−

I

ϕ(t)v(t)dt = −

I

ϕ(t)

]a,t[

dDudt = −

I

ϕ(t)

I

χ]a,t[(s)dDu(s)dt

= −

I

ϕ(t)

I

χ]a,t[(s)dtdDu(s)= −

I

_b

s

ϕ(t)dtdDu(s)

=

Iϕ(s)dDu(s).

Hence we have proved that Dv=Du. Thereforeu−visL¹-a.e. equal to a constant u_a ∈

H

^{thus V}e(u,I)=V_p(u_a+v,I)=V_p(v,I) <∞.

In the same way we can prove that settingw(t):=Du(]a,t]),t ∈I, thenwis right-continuous andu(t)=u_a+Du(]a,t])forL¹-a.e.t ∈I. Therefore we infer the following:

Corollary 2.10. Assume that u ∈ L¹(I;

H

). Then u ∈ BV(I;

H

)if and only if V_e(u,I) <∞. In this case, if u_a ∈

H

is the unique vector such that(2.10)holds, the functions u^l,u^r ∈

H

^{I defined by}

u^l(t):=u_a+Du(]a,t[), u^r(t):=u_a+Du(]a,t]), t ∈I, are respectively the left-continuous and the right-continuous representatives of u (with respect toL¹). We have u_a = u^l(a+)= u^r(a+)and we haveV_p(u^l,I) = V_p(u^r,I)=V_e(u,I)= Du.

If not otherwise specified, we understand that a mapping f ∈ BV(I;

H

)is extended to I by setting f(a) := f^r(a+)and f(b):= f^r(b−)(ifaand/orbare finite).

Corollary 2.11. If u,u_n ∈ BV(I;

H

)are such that u_n → u in L¹(I;

H

), then Du ≤lim inf_n→∞Du_n.

Proof. SinceDv = V_e(v,I) = V_e(v^l,I)it is not restrictive to assume that u andu_n are the left-continuous representatives. Let us consider a subsequence of Du_nwhich is convergent toλ ∈

R

and that we do not relabel. There exists a further subsequencen_k such thatu_nk →uL¹-a.e. in I. Redefining everyu_nk on a suitableL¹-null set of Cont(u)we obtain thatu_nk(t)→u(t)for everyt ∈Cont(u), therefore by Corollary 2.5 V_e(u,I)≤λ. The thesis follows.

Thestrict semimetriconBV(I;

H

)is defined as follows:

d_s(u, v):= u−v_L¹_(I;H)− |Du − Dv|, u, v∈BV(I;

H

^{). (2.11)} Ifd_s(u_n,u)→0 we also say thatu_n →u strictly on I. The strict metric induces a natural topology onBV(I;

H

), indeed we have the following:

Proposition 2.12. If u∈BV(I;

H

)then there exists a sequence(u_n)inC^∞(I;

H

) such that u_n→u strictly on I .

The previous proposition is classical if

H

is finite dimensional. In the Ap- pendix we provide a proof for the general case (see Proposition A.2).

Let us also mention the fact that d_s is not complete, this is important if we consider the problem of extending a BV(I;

H

)-valued operator in a continuous manner.

Let us recall that if Du = vL¹andv ∈ L^p(I;

H

), p ∈ [1,∞], then the dis- tributional derivativeuequalsL¹-a.e. the pointwise derivative andu = vL¹-a.e.

in I. Fork ∈

N

^{we defineWk,p(I;}

H

):= {u∈L^p(I;

H

) : u^(k) ∈L^p(I;

H

)}. It is well known that V_p(u,I)=

Iu(t)_Hdt wheneveru∈W^1,1(I;

H

), therefore

W^1,1(I;

H

) ⊆ BV(I;

H

). Moreover f ∈W^1,∞(I;

H

)if and only if its continu- ous representative belong toLip(I;

H

)∩L¹(I;

H

). The standard semimetric on W^1,p(I;

H

)is

u_W^1,p_(I;H):= uL^p₍I;H)+ uL^p₍I;H), u∈W^1,p(I;

H

) (see the appendix of [4] for details).

2.6. Rate independent operators

Now we recall the notion of (vector) rate independent operator. In the last decades, operators of this kind have been extensively studied in the scalar case in several research articles and in the monographs [6, 12, 14, 19, 27]. The vector case has been object of fewer investigations than the scalar case: seee.g.[12] for the finite dimensional case and [14] for the Hilbert case.

Definition 2.13. Assume thatF ⊆ BV(I;

H

). We say thatR: F −→BV(I;

H

) is arate independent operatorif

R(u◦φ)=R(u)◦φ (2.12)

for everyu∈F and everyφ: I −→ Iincreasing and surjective such thatu◦φ∈F.

Notice that in defining φ from I into itself, we allow, e.g., time rescalings that are equal to b ∈

R

on an interval ]t₀,b[ for a certain timet₀ ∈ ]a,b[. Of course the definition makes sense if we extend anyu ∈BV(I;

H

)to I, by setting

f(a):= f^r(a+), f(b):= f^r(b−), foraand/orbfinite.

Definition 2.14. Assume that F ⊆ BV(I;

H

), F =

∅

. We say thatR : F −→

BV(I;

H

)islocally isotoneif for everyc,d∈ I,c<d,

V_e(u,]c,d[)= u(d)−u(c)_H=⇒V_e(R(u),]c,d[)

= R(u)(d)−R(u)(c)_H. (2.13) The notion of locally isotone rate independent operator was introduced in [21, Re- mark 4.6] and it is a natural generalization of the notion of local monotonicity, well known in hysteresis. In the scalar case the local monotonicity ofRmeans that if R(u)is monotone increasing (respectively decreasing) on [c,d] thanuis monotone increasing (respectively decreasing) on the same interval. Instead condition (2.13) simply means thatR(u)is monotone on ]c,d[ wheneveruis monotone on [c,d], hence the term ‘isotone’. Since we will use Definition 2.14 only for F⊆C(J;

H

), the essential variation can be replaced by the pointwise variation on [c,d]. In this case and when the dimension of

H

is greater than one, condition (2.13) means that if u is an injective parametrization of a segment on [c,d], thenR(u) is also an injective parametrization of another segment on [c,d].

3.

Main results

In this section we state the main results of this paper. To this aim we first need some properties on reparametrizations. We setI =]a,b[ witha,b∈

R

^,a<b.

3.1. Reparametrizations

We follow [11, Section 2.5.16], with some slight differences, due to the fact that we assign the same arc length to two functions which are equalL¹-a.e. Moreover we need a normalization factor. Set I := ]a,b[ witha,b ∈

R

^{, a} < b. Ifu ∈ BV(I;

H

^{), defineu :[a,b]−→[a,b] by}

u(t):=

a+ ^b_Du^−aDu(]a,t[) ifDu =0

a ifDu =0, t ∈I. (3.1)

The functionu is increasing and left-continuous. Moreover Discont(u) = A(u) and

u(I)= I

t∈A(u)

]u(t), u(t+)].

If t₁ < t₂ we have u^l(t₁)− u^l(t₂)_H ≤ Du([t₁,t₂[) = Du(]a,t₂[)−



Du(]a,t₁[)therefore

u^l(t₁)−u^l(t₂)_H≤ Du

b−a |u(t₁)−u(t₂)| ∀t₁,t₂∈ I. (3.2) This inequality yields thatu^l(⁻_u¹(σ))is a singleton for everyσ ∈u(I), therefore there is a unique functionU :u(I)−→

H

^{such thatU}◦=u^l. From (3.2) it also follows thatU is the unique Lipschitz function such thatU◦u=uL¹-a.e. and its Lipschitz constant satisfies Lip(U)≤ Du/(b−a). In order to extendU to all ofI we defineu: I −→

H

^{by setting}

u(σ ):=(1−λ)u^l(t)+λu^l(t+)ifσ=(1−λ)u(t)+λu(t+),t ∈I,λ∈ [0,1].

It is clear thatu extendsU and that Lip(u) = Lip(U). The functionu may be regarded as a kind of reparametrization of u by the normalized arc length. We summarize the previous discussions in the following proposition.

Proposition 3.1. Assume a,b are finite and let u ∈BV(I;

H

). Letu : I −→ I be its “normalized” arc length defined by(3.1). Then there exists a unique function u∈Lip(I;

H

)such that

u=u◦u L¹-a.e. in I, (3.3) u is affine on [u(t), u(t+)] ∀t ∈ A(u). (3.4)

3.2. Main abstract results Here is our main result.

Theorem 3.2. Let I be bounded. Assume thatR : Lip(I;

H

) −→ BV(I;

H

)∩ C(I;

H

)is a rate independent operator which is continuous whenLip(I;

H

)and BV(I;

H

)∩C(I;

H

)are endowed with the strict topology. ThenRadmits a unique continuous extension to BV(I;

H

)∩ C(I;

H

). Moreover R can be continuosly extended to all of BV(I;

H

) if and only if R is locally isotone. This extension is unique if we identify functions which are L¹-a.e. equal in I and it is given by R:BV(I;

H

)−→BV(I;

H

)

R(u):=R(u)◦u, u∈BV(I;

H

), (3.5) whereu anduare defined by Proposition3.1. The operatorRis rate independent.

Even ifRis not locally isotone we have the following continuity property Proposition 3.3. Let I be bounded. Assume thatR:Lip(I;

H

)−→BV(I;

H

)∩ C(I;

H

)is a rate independent operator which is continuous whenLip(I;

H

)and BV(I;

H

)∩C(I;

H

)are endowed with the strict topology. LetR:BV(I;

H

)−→

BV(I;

H

⁾be defined by formula(3.5). ThenR(u_n)−R(u)_L¹_(I;H) →0whenever u_n →u strictly on I , u,u_n ∈BV(I;

H

).

Finally we present the following theorem that will allows us to infer new con- tinuity properties of the vector play operator (defined in Section 3.3) also in the classical framework of absolutely continuous inputs.

Theorem 3.4. Let I be bounded. LetFbe such thatLip(I;

H

^{)⊆F⊆BV(I;}

H

^)∩

C(I;

H

). Assume that R : F −→ BV(I;

H

)∩C(I;

H

)is rate independent and has the following continuity property:

v,vn∈Lip(I;

H

), vn−v_W^1,1_(I;H)→0=⇒R(vn)→R(v)strictly on I (3.6) as n→ ∞. ThenRis continuous with respect to the strict topology, i.e.

u_n →u strictly on I =⇒ R(u_n)→R(u) strictly on I (3.7) as n→ ∞.

The previous theorem implies in particular that Theorem 3.2 holds if we re- place the strict continuity by the condition (3.6), which is well-known in many particular concrete cases.

Remark 3.5. We point out that we proved a particular case of Theorem 3.2 in [21, 22]: namely the case

H

=

R

, even if in those papers we did not observe that the existence of the continuous extension toBV(I;

R

^)∩C(I;

R

⁾is granted even ifR is not locally isotone. The scalar version of Theorem 3.4 is proved in [24].

The vectorial case is not a rephrasing of the scalar case, but different proofs are needed. Moreover in the vector case the condition of local isotonicity has a clear geometrical meaning. This kind of geodesic condition allows to infer new continuity properties of the vector play operator that are very different form the scalar case. This analysis is performed in Section 5.

3.3. Main applications

In this section we state the main applications of the abstract theorems to rate inde- pendent variational inequalities. We assume that

Z

is a closed convex subset of

H

, 0∈

Z

, (3.8)

z₀∈

Z

, (3.9)

0<T <∞. (3.10)

In order to define the play operator we need to recall the following result.

Proposition 3.6. For every u∈W^1,∞(]0,T[;

H

)there exists y∈W^1,∞(]0,T[;

H

) such that

u(t)−y(t)∈

Z

^{forL1-a.e. t}∈ ]0,T[, (3.11) u(t)−y(t)−z,y(t) ≥0 ∀z∈

Z

, forL¹-a.e. t ∈]0,T[, (3.12)

u(0)−y(0)=z₀. (3.13)

There is a unique y ∈ C([0,T];

H

) which satisfies (3.11)-(3.13) (equivalently such solution is unique if we identify functions agreeing outside a set having zero Lebesgue measure).

The previous result is well-known, anyway we will need to outline its proof in Section 5.1. If u ∈ W^1,1(]0,T[;

H

) andP(u) := y, where y is the unique continuous solution of (3.11)−(3.13), we define an operator

P:W^1,1(]0,T[;

H

^{)−→W1,1(]0,T[;}

H

⁾

which is usually called (vector)play operator. It is well known thatPis rate inde- pendent. The main application of the abstract results is the following

Theorem 3.7. The play operator is continuous with respect to the strict topology and it admits a unique continuous extension to BV(]0,T[;

H

)∩C([0,T];

H

). Moreover it can be continuously extended toBV(]0,T[;

H

)if and only if

Z

^{is a} vector subspace or if

Z

= {x ∈

H

: −α≤ f,x ≤β}

for some f ∈

H

^{{0}andα, β∈[0,∞}]. In both cases such extension is given by P(u)=P(u)◦u,

whereu ∈ Lip([0,T];

H

) and u are defined by Proposition 3.1 with a = 0, b=T .

Let us observe that the Theorem 3.7 improves a previous result in [14, Proposi- tion 4.11], where the continuity ofPinBV(]0,T[;

H

)∩C([0,T];

H

)was proved for separable

H

^{and for}

Z

having suitable regularity properties,i.e. such that at every point x ∈ ∂

Z

there exists a unique outward normaln(x)and the resulting mappingnis continuous (see the Appendix A.5 for the notion of normal vectors).

Moreover we answer in a complete manner to the open question about the continuous extendibility of the play operator toBV(]0,T[;

H

^).

Remark 3.8. Let us remark that in [14] the strict metric is defined byd˜_s(u, v):=

u−v_∞+|V_p(u,[0,T])−V_p(u,[0,T])|foru, v∈BV(]0,T[;

H

^)∩C([0,T];

H

⁾ continuous of bounded variation. But in the continuous case this turns out to be topologically equivalent to the definition adopted in our paper, by virtue of Corol- lary 4.8 of Section 4.2 below.

4.

Proof of abstract results

Let us recall thatI :=]a,b[, witha,b∈[−∞,∞],a<b.

4.1. Properties of reparametrizions

Lemma 4.1. Letv: I −→

H

be such thatV_p(v,I) <∞and letβ: I −→ I be an increasing function satisfyingβ(a)=a,β(b)=b, andDiscont(v)∩Discont(β)=

∅

. Moreover assume that

V_p(v,[β(t−), β(t+)])= v(β(t+))−v(β(t−))_H ∀t ∈Discont(β). (4.1) ThenV_p(v◦β,I)=V_p(v,I).

Proof. We prove the lemma whenβ is left-continuous andβ(a) = β(a+)(fora finite), the other cases being similar (however we need only this case). The in- equality V_p(v◦β,I)≤V_p(v,I)is obvious, hence V_p(v,I)is an upper bound for {_n

j=1v(β(t_j))−v(β(t_j−1))_H : n∈

N

^{, a<t0≤ · · · ≤tn<b}. Letε >0 be} arbitrarily fixed. There exists a subdivision(t_j)ⁿ_j=0ofI such that

V_p(v,I) <ⁿ

j=1

v(t_j)−v(t_j−1)_H+ε/2. (4.2) For everyσ ∈ Discont(β)there is a possibly empty subset E_σ ⊆ {t_j}contained in [β(σ−), β(σ+)[. Adding the points β(σ−) = β(σ ), β(σ+)to E_σ, the sum in (4.2) can only increase. Moreover, thanks to the assumption (4.1) we can also replace E_σ by {β(σ), β(σ+)}without affecting such a sum. Therefore we can assume that (4.2) holds for a subdivision(t_j)such that

{t_j}ⁿ_j=0={s₀¹, . . . ,s_k¹

1−1} ∪ {β(σ1),β(σ1+)} ∪ {s₀², . . . ,s_k²

2−1} ∪ {β(σ2),β(σ2+)}∪

· · · ∪ {s₀^m, . . . ,s_k^m

m−1} ∪ {β(σm), β(σm+)} ∪ {s^m₀⁺¹, . . . ,s_k^m+1

m+1}

where

σi ∈Discont(β), s_kⁱ_m :=β(σi) ∀i =1, . . . ,m; {s₀ⁱ, . . . ,s_kⁱ_i} ⊆β(I) ∀i =1, . . . ,m+1.

Hence, setting

τ_hⁱ :=β⁻¹(s_hⁱ) i =1, . . . ,m+1, j =0, . . . ,k_m+1, we can write (βis left-continuous)

n j=1

v(t_j)−v(t_j−1)_H

= m i=1

ki h=1

v(β(τ_hⁱ))−v(β(τ_hⁱ₋₁))_H+ v(β(σi+))−v(β(σi))_H

+ v(β(τ₀ⁱ⁺¹))−v(β(σi+))_H

+

k_m+1

h=1

v(β(τ_h^m+1))−v(β(τ_h^m₋⁺₁¹))_H

The fact that Discont(v)∩Discont(β)=

∅

yields that for everyi =1, . . . ,mthere existsσ˜i very nearσi, such thatσi <σ˜i andv(β(σi+))−v(β(σ˜i))_H< ε/(2m), so that

n j=1

v(t_j)−v(t_j−1)_H

≤ m i=1

ki h=1

v(β(τ_hⁱ)))−v(β(τ_hⁱ₋₁)_H+ v(β(σ˜i))−v(β(σi))_H

+ v(β(τ₀ⁱ⁺¹))−v(β(σ˜i))_H+ε/m

+

k_m+1

h=1

v(β(τ_h^m+1))−v(β(τ_h^m₋⁺₁¹))_H.

That is, we have found a subdivision(θj)^r_j=0such that V_p(v,I)<_n

j=1v(β(θj))−

v(β(θj−1))_H+ε,and the lemma is proved.

Lemma 4.2. Assume that I is bounded and that u ∈BV(I;

H

). Letu andu be the maps provided by Proposition3.1. Letφ :I −→I be increasing and surjective, and setv:=u◦φ. Then_v =u◦φandv=u◦_v, or in other termsu

◦φ =u.